On the ε-entropy of diffusion processes

Summary

In the present paper we give an evaluation of ε-entropy of a one dimensional diffusion process ξ(t) 0≦t≦T whose generator is

$$\mathfrak{G} = \frac{1}{2}a(x)^2 \frac{{d^2 }}{{dx^2 }} (x \in R),$$

((1))

where the diffusion coefficienta(x) ² satisfies

$$\left| {a(x) - a(y)} \right| \leqq L\left| {x - y} \right|,$$

((2))

$$0< k \leqq a(x)^2 \leqq K$$

((3))

for everyx andy (L, k andK are positive constants). We assume ξ(0)=0 for simplicity of calculation. Then we can prove the following:Theorem.Under the conditions (2)and (3),ε-entropy H _ε ({ξ(t)})of ξ(t), 0≦t≦T is asymptotically evaluated for small ε>0

$$\frac{k}{{eK}} \cdot \frac{1}{{\pi ^2 }}\left\{ {\int_0^T {\sqrt {Ea(\xi (u))^2 } du} } \right\}^2 \frac{1}{{\varepsilon ^2 }} \precsim H_\varepsilon (\left\{ {\xi (t)} \right\}) \precsim \frac{2}{{\pi ^2 }}\left\{ {\int_0^T {\sqrt {Ea(\xi (u))^2 } du} } \right\}^2 \frac{1}{{\varepsilon ^2 }}.$$

Previously Kolmogorov stated in [3] without proof “For a diffusion process whose generator is given by (1)H _ε ({ξ(t)}) is calculated by the formula:

$$H_\varepsilon (\left\{ {\xi (t)} \right\}) = \frac{4}{\pi }\left\{ {\int_0^T {Ea(\xi (u))^2 du} } \right\}\frac{1}{{\varepsilon ^2 }} + o\left( {\frac{1}{{\varepsilon ^2 }}} \right) (\varepsilon \to 0)$$

under certain natural conditions”. However, in consideration of Pinsker’s results for Gaussian processes [5] and our present theorem this formula appears inaccurate. For the proof of the theorem we use the well-known formula of ε-entropy for finite dimensional random variables (lemma 3).